Transposition bijects even and odd permutations

Dependencies:

1. Parity of a permutation

Let $\lambda_\tau(\sigma) = \tau\sigma$, where $\tau$ is a transposition.

$\lambda_\tau$ is a bijection from the set of even permutations to the set of odd permutations.

Proof

$ \textrm{Let } \lambda_\tau(\sigma_1) = \lambda_\tau(\sigma_2) \implies \tau\sigma_1 = \tau\sigma_2 \implies \sigma_1 = \tau^{-1}\tau\sigma_2 = \sigma_2 $

Therefore, $\lambda_\tau$ is one-to-one.

Let $\mu$ be an odd permutation. Then $\lambda_\tau(\mu) = \tau\mu$ is an even permutation.

$\lambda_\tau(\lambda_\tau(\mu)) = \tau\tau\mu = \mu$. So $\mu$ has an even permutation as a pre-image in $\lambda_\tau$. Therefore, $\lambda_\tau$ is onto.

Therefore, $\lambda_\tau$ is a bijection from even permutations to odd permutations.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 10

Transitive dependencies:

1. /sets-and-relations/relation-composition-is-associative
2. /sets-and-relations/composition-of-bijections-is-a-bijection
3. Group
4. Subgroup
5. Permutation group
6. Product of cycles and a transposition
7. Permutation is disjoint cycle product
8. Product of disjoint cycles is commutative
9. Canonical cycle notation of a permutation
10. Parity of a permutation